Learn English – Should I hyphenate “up-to-the-minute” in this case

You would not use hyphens when the phrase is used as a noun:

Blood red had always been his favorite color.
The flag had white lion on a field of forest green.
The dress was a fine royal blue.

Generally, I'd recommend that you use hyphens when the phrase is used as an adjective (i.e. a compound modifier):

He drank the sweet, blood-red wine.
The flag, forest-green and proud, was raised at noon.
The royal-blue dress fitted her well.

However, if the color phrase is part of a larger adjectival phrase where the color phrase itself is being modified, leave the hyphens off:

Her extravagantly royal blue dress was a sight to behold.

There may be still be some disagreement on this, though, and there are no absolute rules that apply to all possible word combinations. You are free to use whatever form you feel is most natural and easy to understand.

You are right on both counts and I like your version of the sentence the best. There is no need for "30-minute" but it is acceptable and "15- to 30-minutes" is a fine suggestion by @FumbleFingers. I used thepunctuationguide.com as my reference.